Average Error: 17.9 → 0.2
Time: 12.5s
Precision: 64
\[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.9931203730725725042915996709780301898718:\\ \;\;\;\;\log \left(\sqrt{e^{1}}\right) + \log \left(\frac{\sqrt{e^{1}}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{e^{1}}\right) + \log \left(\frac{\sqrt{e^{1}}}{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}}\right)\\ \end{array}\]
1 - \log \left(1 - \frac{x - y}{1 - y}\right)
↓
\begin{array}{l}
\mathbf{if}\;\frac{x - y}{1 - y} \le 0.9931203730725725042915996709780301898718:\\
\;\;\;\;\log \left(\sqrt{e^{1}}\right) + \log \left(\frac{\sqrt{e^{1}}}{1 - \frac{x - y}{1 - y}}\right)\\

\mathbf{else}:\\
\;\;\;\;\log \left(\sqrt{e^{1}}\right) + \log \left(\frac{\sqrt{e^{1}}}{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}}\right)\\

\end{array}
double f(double x, double y) {
        double r278959 = 1.0;
        double r278960 = x;
        double r278961 = y;
        double r278962 = r278960 - r278961;
        double r278963 = r278959 - r278961;
        double r278964 = r278962 / r278963;
        double r278965 = r278959 - r278964;
        double r278966 = log(r278965);
        double r278967 = r278959 - r278966;
        return r278967;
}
↓
double f(double x, double y) {
        double r278968 = x;
        double r278969 = y;
        double r278970 = r278968 - r278969;
        double r278971 = 1.0;
        double r278972 = r278971 - r278969;
        double r278973 = r278970 / r278972;
        double r278974 = 0.9931203730725725;
        bool r278975 = r278973 <= r278974;
        double r278976 = exp(r278971);
        double r278977 = sqrt(r278976);
        double r278978 = log(r278977);
        double r278979 = r278971 - r278973;
        double r278980 = r278977 / r278979;
        double r278981 = log(r278980);
        double r278982 = r278978 + r278981;
        double r278983 = 2.0;
        double r278984 = pow(r278969, r278983);
        double r278985 = r278968 / r278984;
        double r278986 = 1.0;
        double r278987 = r278986 / r278969;
        double r278988 = r278985 - r278987;
        double r278989 = r278971 * r278988;
        double r278990 = r278968 / r278969;
        double r278991 = r278989 + r278990;
        double r278992 = r278977 / r278991;
        double r278993 = log(r278992);
        double r278994 = r278978 + r278993;
        double r278995 = r278975 ? r278982 : r278994;
        return r278995;
}

Error

Bits error versus x

Bits error versus y

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Target

Original17.9
Target0.1
Herbie0.2
\[\begin{array}{l} \mathbf{if}\;y \lt -81284752.6194724142551422119140625:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \mathbf{elif}\;y \lt 30094271212461763678175232:\\ \;\;\;\;\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;1 - \log \left(\frac{x}{y \cdot y} - \left(\frac{1}{y} - \frac{x}{y}\right)\right)\\ \end{array}\]

Derivation

  1. Split input into 2 regimes

  2. if (/ (- x y) (- 1.0 y)) < 0.9931203730725725

    1. Initial program 0.0

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
    2. Using strategy rm

    3. Applied add-log-exp0.0

      \[\leadsto \color{blue}{\log \left(e^{1}\right)} - \log \left(1 - \frac{x - y}{1 - y}\right)\]

    4. Applied diff-log0.0

      \[\leadsto \color{blue}{\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity0.0

      \[\leadsto \log \left(\frac{e^{1}}{\color{blue}{1 \cdot \left(1 - \frac{x - y}{1 - y}\right)}}\right)\]

    7. Applied add-sqr-sqrt0.0

      \[\leadsto \log \left(\frac{\color{blue}{\sqrt{e^{1}} \cdot \sqrt{e^{1}}}}{1 \cdot \left(1 - \frac{x - y}{1 - y}\right)}\right)\]

    8. Applied times-frac0.0

      \[\leadsto \log \color{blue}{\left(\frac{\sqrt{e^{1}}}{1} \cdot \frac{\sqrt{e^{1}}}{1 - \frac{x - y}{1 - y}}\right)}\]

    9. Applied log-prod0.0

      \[\leadsto \color{blue}{\log \left(\frac{\sqrt{e^{1}}}{1}\right) + \log \left(\frac{\sqrt{e^{1}}}{1 - \frac{x - y}{1 - y}}\right)}\]

    10. Simplified0.0

      \[\leadsto \color{blue}{\log \left(\sqrt{e^{1}}\right)} + \log \left(\frac{\sqrt{e^{1}}}{1 - \frac{x - y}{1 - y}}\right)\]

    if 0.9931203730725725 < (/ (- x y) (- 1.0 y))

    1. Initial program 61.5

      \[1 - \log \left(1 - \frac{x - y}{1 - y}\right)\]
    2. Using strategy rm

    3. Applied add-log-exp61.5

      \[\leadsto \color{blue}{\log \left(e^{1}\right)} - \log \left(1 - \frac{x - y}{1 - y}\right)\]

    4. Applied diff-log61.5

      \[\leadsto \color{blue}{\log \left(\frac{e^{1}}{1 - \frac{x - y}{1 - y}}\right)}\]
    5. Using strategy rm

    6. Applied *-un-lft-identity61.5

      \[\leadsto \log \left(\frac{e^{1}}{\color{blue}{1 \cdot \left(1 - \frac{x - y}{1 - y}\right)}}\right)\]

    7. Applied add-sqr-sqrt61.5

      \[\leadsto \log \left(\frac{\color{blue}{\sqrt{e^{1}} \cdot \sqrt{e^{1}}}}{1 \cdot \left(1 - \frac{x - y}{1 - y}\right)}\right)\]

    8. Applied times-frac61.5

      \[\leadsto \log \color{blue}{\left(\frac{\sqrt{e^{1}}}{1} \cdot \frac{\sqrt{e^{1}}}{1 - \frac{x - y}{1 - y}}\right)}\]

    9. Applied log-prod61.5

      \[\leadsto \color{blue}{\log \left(\frac{\sqrt{e^{1}}}{1}\right) + \log \left(\frac{\sqrt{e^{1}}}{1 - \frac{x - y}{1 - y}}\right)}\]

    10. Simplified61.5

      \[\leadsto \color{blue}{\log \left(\sqrt{e^{1}}\right)} + \log \left(\frac{\sqrt{e^{1}}}{1 - \frac{x - y}{1 - y}}\right)\]

    11. Taylor expanded around inf 0.6

      \[\leadsto \log \left(\sqrt{e^{1}}\right) + \log \left(\frac{\sqrt{e^{1}}}{\color{blue}{\left(\frac{x}{y} + 1 \cdot \frac{x}{{y}^{2}}\right) - 1 \cdot \frac{1}{y}}}\right)\]

    12. Simplified0.6

      \[\leadsto \log \left(\sqrt{e^{1}}\right) + \log \left(\frac{\sqrt{e^{1}}}{\color{blue}{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}}}\right)\]
  3. Recombined 2 regimes into one program.

  4. Final simplification0.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{1 - y} \le 0.9931203730725725042915996709780301898718:\\ \;\;\;\;\log \left(\sqrt{e^{1}}\right) + \log \left(\frac{\sqrt{e^{1}}}{1 - \frac{x - y}{1 - y}}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(\sqrt{e^{1}}\right) + \log \left(\frac{\sqrt{e^{1}}}{1 \cdot \left(\frac{x}{{y}^{2}} - \frac{1}{y}\right) + \frac{x}{y}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2019297 
(FPCore (x y)
  :name "Numeric.SpecFunctions:invIncompleteGamma from math-functions-0.1.5.2, B"
  :precision binary64

  :herbie-target
  (if (< y -81284752.619472414) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y))))) (if (< y 3.0094271212461764e25) (log (/ (exp 1) (- 1 (/ (- x y) (- 1 y))))) (- 1 (log (- (/ x (* y y)) (- (/ 1 y) (/ x y)))))))

  (- 1 (log (- 1 (/ (- x y) (- 1 y))))))